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Optimal Level of Product Availability

Chapter 12 of Chopra

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Outline Determining optimal level of product availability

– Single order in a season

– Continuously stocked items

Ordering under capacity constraints Levers to improve supply chain profitability

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Motivating News Article:

Mattel, Inc. & Toys “R” UsMattel [who introduced Barbie in 1959 and run a stock out for several years then on] was hurt last year by inventory cutbacks at Toys “R” Us, and officials are also eager to avoid a repeat of the 1998 Thanksgiving weekend. Mattel had expected to ship a lot of merchandise after the weekend, but retailers, wary of excess inventory, stopped ordering from Mattel. That led the company to report a $500 million sales shortfall in the last weeks of the year ... For the crucial holiday selling season this year, Mattel said it will require retailers to place their full orders before Thanksgiving. And, for the first time, the company will no longer take reorders in December, Ms. Barad said. This will enable Mattel to tailor production more closely to demand and avoid building inventory for orders that don't come.

- Wall Street Journal, Feb. 18, 1999

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Key Questions

How much should Toys R Us order given demand uncertainty?

How much should Mattel order? Will Mattel’s action help or hurt profitability? What actions can improve supply chain profitability?

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Another Example: Apparel IndustryHow much to order? Parkas at L.L. Bean

Demand D_i

Probabability p_i

Cumulative Probability of demand being this size or less, F(.)

Probability of demand greater than this size, 1-F(.)

4 .01 .01 .99 5 .02 .03 .97 6 .04 .07 .93 7 .08 .15 .85 8 .09 .24 .76 9 .11 .35 .65

10 .16 .51 .49 11 .20 .71 .29 12 .11 .82 .18 13 .10 .92 .08 14 .04 .96 .04 15 .02 .98 .02 16 .01 .99 .01 17 .01 1.00 .00

Expected demand is 1,026 parkas.

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Parkas at L.L. Bean

Cost per parka = c = $45

Sale price per parka = p = $100

Discount price per parka = $50

Holding and transportation cost = $10

Salvage value per parka = s = 50-10=$40

Profit from selling parka = p-c = 100-45 = $55

Cost of overstocking = c-s = 45-40 = $5

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Optimal level of product availability

p = sale price; s = outlet or salvage price; c = purchase price

CSL = Probability that demand will be at or below reorder point

Raising the order size if the order size is already optimalExpected Marginal Benefit =

=P(Demand is above stock)*(Profit from sales)=(1-CSL)(p - c)

Expected Marginal Cost =

=P(Demand is below stock)*(Loss from discounting)=CSL(c - s)

Define Co= c-s; Cu=p-c

(1-CSL)Cu = CSL Co

CSL= Cu / (Cu + Co)

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Order Quantity for a Single Order

Co = Cost of overstocking = $5

Cu = Cost of understocking = $55

Q* = Optimal order size

917.0555

55)( *

ou

u

CC

CQDemandPCSL

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Optimal Order Quantity

0

0.2

0.4

0.6

0.8

1

1.2

4 5 6 7 8 9 10 11 12 13 14 15 16 87

CumulativeProbability

Optimal Order Quantity = 13(‘00)

0.917

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Parkas at L.L. Bean

Expected demand = 10 (‘00) parkas Expected profit from ordering 10 (‘00) parkas = $499

Approximate Expected profit from ordering 1(‘00) extra parkas if 10(’00) are already ordered

= 100.55.P(D>=1100) - 100.5.P(D<1100)

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Parkas at L.L. Bean

Additional100s

ExpectedMarginal Benefit

ExpectedMarginal Cost

Expected MarginalContribution

11th 5500.49 = 2695 500.51 = 255 2695-255 = 2440

12th 5500.29 = 1595 500.71 = 355 1595-355 = 1240

13th 5500.18 = 990 500.82 = 410 990-410 = 580

14th 5500.08 = 440 500.92 = 460 440-460 = -20

15th 5500.04 = 220 500.96 = 480 220-480 = -260

16th 5500.02 = 110 500.98 = 490 110-490 = -380

17th 5500.01 = 55 500.99 = 495 55-495 = -440

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Revisit L.L. Bean as a Newsvendor Problem Total cost by ordering Q units:

– C(Q) = overstocking cost + understocking cost

Qu

Q

o dxxfQxCdxxfxQCQC )()()()()(0

0))(())(1()()(

uuouo CCCQFQFCQFCdQ

QdC

Marginal cost of raising Q* - Marginal cost of decreasing Q* = 0

uo

u

CC

CQDPQF

)()( **

Show Excel to compute expected single-period cost curve.

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Ordering Women’s Designer Boots Under Capacity Constraints

Autumn Leaves Ruffle Retail price $150 $200 $250

Purchase price $75 $90 $110 Salvage price $40 $50 $90 Mean Demand 1000 500 250

Stand. deviation of demand 250 175 125

Available Store Capacity = 1,500.

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Assuming No Capacity Constraints

Autumn Leaves Ruffle pi-ci 150-75=$75 200-90=$110 250-110=$140 ci-si 75-40=$35 90 - 50 = $40 110-90 = $20

Critical Fractile 75/110 = 0.68 110/150= 0.73 140/160=0.875 zi 0.47 0.61 1.15 Qi 1118 607 394

Storage capacity is not sufficient to keep all models!

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Algorithm for Ordering Under Capacity Constraints

{Initialization}

ForAll products, Qi := 0. Remaining_capacity:=Total_capacity.

{Iterative step}While Remaining_capacity > 0 do

ForAll products,

Compute the marginal contribution of increasing Qi by 1If all marginal contributions <=0, STOP{Order sizes are already sufficiently large for all products}else Find the product with the largest marginal contribution, call it j

{Priority given to the most profitable product}

Qj := Qj+1 and Remaining_capacity=Remaining_capacity-1

{Order more of the most profitable product}

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Marginal Contribution=(p-c)P(D>Q)-(c-s)P(D<Q) Order Quantity Marginal Contribution

Remaining_Capacity Autumn Leaves Ruffle Autumn Leaves Ruffle 1500 0 0 0 74.997 109.679 136.360 1490 0 0 10 74.997 109.679 135.611

1360 0 0 140 74.997 109.679 109.691 1350 0 0 150 74.997 109.679 106.103 1340 0 10 150 74.997 109.617 106.103 1330 0 20 150 74.997 109.543 106.103 1320 0 30 150 74.997 109.457 106.103 1310 0 40 150 74.997 109.357 106.103

890 0 380 230 74.997 73.033 70.170 880 10 380 230 74.996 73.033 70.170 870 20 380 230 74.995 73.033 70.170

290 580 400 230 69.887 67.422 70.170 280 580 400 240 69.887 67.422 65.101

1 788 446 265 53.196 53.176 52.359 0 789 446 265 53.073 53.176 52.359

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Optimal Safety Inventory and Order Levels:(ROP,Q) ordering model

Lead Times

time

inventory

Shortage

An inventory cycle

ROP

Q

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A Cost minimization approach as opposed to the last chapter’s service based approach

Fixed ordering cost = S R / Q Holding cost = h C (Q/2+ss) where ss = ROP – L R Backordering cost (based on per unit backordered),

with f(.), the distribution of the demand during the lead time,

Total cost per time

ROP

dxxfROPxbQ

R)()(

per time

CostBackorder

ROP

per timeCostHolding

timeperCostOrdering

dxxfROPxQ

RbLRROP

QhC

Q

RS

)()(}2

{

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Optimal Q (for high service level) and ROP

Q*=Optimal lot size ROP*=Optimal reorder point

A cost / benefit analysis to obtain CSL:– (1-CSL)bR/Q= per time benefit of increasing ROP by 1

» (1-CSL)b= per cycle benefit of increasing ROP by 1

– hC= per time cost of increasing ROP by 1– (1-CSL)bR/Q=hC gives the optimality equation for ROP

hC

SRQ

2* Rb

hCQROPFCSL 1)( **

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Imputed Cost of Backordering

R = 100 gallons/week; R= 20; H=hC= $0.6/gal./year

L = 2 weeks; Q = 400; ROP = 300.

What is the “imputed cost” of backordering?

Let us use a week as time unit. H=0.6/52 per gal per week. Recall the formula

CSL = 1-HQ*/bR

per weekgallon per 8.230$100*0002.0

400*)52/6.0(

)1(

9998.0)1,2.28,200,300( ),,(

RCSL

HQb

normdistLRLROPFCSL R

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Levers for Increasing Supply Chain Profitability Increase salvage value

» Obermeyer sells winter clothing in south America during the summer.» Sell the Xmas trees to Orthodox Christians after Xmas.» Buyback contracts, to be discussed.

Decrease the margin lost from a stock out– Pooling:

» Between the retailers of the same company. Ex. Volvo trucks.

» Between franchises/competitors. Franchises: Car part suppliers, McMaster-Carr and Grainger, are competitors but they buy from

each other to satisfy the customer demand during a stock out. Competitors: BMW dealers in the metroplex: Richardson, Dallas, Arlington, Forth Worth

– Dallas competes with Richardson so no pooling between them– Dallas pools inventory with the rest– Transportation cost of pooling a car from another dealer $1,500– Rebalancing: No transportation cost if cars are switched in the ship in the Atlantic

Improve forecasting to lower uncertainty Quick response by decreasing replenishment lead time which leads to a larger number of orders per

season Postponement of product differentiation Tailored (dual) sourcing

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Impact of Improving ForecastsEX: Demand is Normally distributed with a mean of R = 350 and standard

deviation of R = 150Purchase price = $100 , Retail price = $250Disposal value = $85 , Holding cost for season = $5

How many units should be ordered as R changes?

Price=p=250; Salvage value=s=85-5=80; Cost=c=100

Understocking cost=p-c=250-100=$150, Overstocking cost=c-s=100-80=$20Critical ratio=150/(150+20)=0.88Optimal order quantity=Norminv(0.88,350,150)=526 units

Expected profit? Expected profit differs from the expected cost by a constant.

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Computing the Expected Profit with Normal Demands

σ,1))μ,,normdist(Q(1 Q c)(p

σ,1)μ,,normdist(Q Q s)-(c -

)μ)/σ,0,1,0-Qnormdist(( σ s)-(p -

)μ)/σ,0,1,1-Qnormdist((μ s)-(p Profit Expected

σdeviation standard andμ mean with Normal is demand that theSuppose

dx f(x) Q)Profit(x,Profit Expected

Q xif c)Q-(p

Q xif x)-s)(Q-(c-c)x-(pQ)Profit(x,

quantity.order :Q f(x); pdf with demand :x

unit.per cost):c value;salvage :s price;:(p

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Impact of Improving Forecasts

R Q* Expected

Overstock Expected

Understock Expected

Profit 150 526 186.7 8.6 $47,469

120 491 149.3 6.9 $48,476

90 456 112.0 5.2 $49,482

60 420 74.7 3.5 $50,488

30 385 37.3 1.7 $51,494

0 350 0 0 $52,500

Where is the trade off? Expected overstock vs. Expected understock. Expected profit vs. ?????

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Cost or Profit; Does it matter?

optimal. same theyield they ;equivalent are Q)]E[Cost(x,Min and Q)],E[Profit(xMax

Qin Constant )c)E(Demand-(pQ)]E[Cost(x,Q)],E[Profit(x

c)x(pQ xifc)x -(p

Q xifc)x -(pQ)Cost(x,Q)Profit(x,

Q xif Q)-c)(x-(p

Q xif x)-s)(Q-(cQ)Cost(x,

Q xif c)Q-(p

Q xif x)-s)(Q-(c-c)x-(pQ)Profit(x,

quantity.order :Q f(x); pdf with demand :x

unit.per cost):c value;salvage :s price;:(p

QQ

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Quick Response: Multiple Orders per Season

Ordering shawls at a department store– Selling season = 14 weeks (from 1 Oct to 1 Jan)– Cost per shawl = $40– Sale price = $150– Disposal price = $30– Holding cost = $2 per week

Expected weekly demand = 20 StDev of weekly demand = 15

Understocking cost=150-40=$110 per shawl Overstocking cost=40-30+(14)2=$38 per shawl Critical ratio=110/(110+38)=0.743=CSL

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Ordering Twice as Opposed to Once

The second order can be used to correct the demand supply mismatch in the first order– At the time of placing the second order, take out the on-

hand inventory from the demand the second order is supposed to satisfy. This is a simple inventory correction idea.

Between the times the first and the second orders are placed, more information becomes available to demand forecasters. The second order is typically made against less uncertainty than the first order is.

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Impact of Quick ResponseCorrecting the mismatch with the second order

Single Order Two Orders in a Season Service Level CSL

Order Size

Ending Invent.

Expect. Profit

Initial Order

OUL for 2nd Order

Ending Invent.

Average Total Order

Expect. Profit

0.96 378 97 $23,624 209 209 69 349 $26,590

0.94 367 86 $24,034 201 201 60 342 $27,085

0.91 355 73 $24,617 193 193 52 332 $27,154

0.87 343 66 $24,386 184 184 43 319 $26,944

0.81 329 55 $24,609 174 174 36 313 $27,413

0.75 317 41 $25,205 166 166 32 302 $26,916

OUL: Ideal Order Up to Level of inventory at the beginning of a cycle

Average total order approximately = OUL1+OUL2-Ending Inventory

As we decrease CSL, profit first increases, then decreases and finally increases again. The profits are computed via simulation.

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Forecasts Improve for the Second Order Uncertainty reduction from SD=15 to 3

Single Order Two Orders in a Season Service Level

Order Size

Ending Invent.

Expect. Profit

Initial Order

OUL for 2nd Order

Average Total Order

Ending Invent.

Expect. Profit

0.96 378 96 $23,707 209 153 292 19 $27,007

0.94 367 84 $24,303 201 152 293 18 $27,371

0.91 355 76 $24,154 193 150 288 17 $26,946

0.87 343 63 $24,807 184 148 288 14 $27,583

0.81 329 52 $24,998 174 146 283 14 $27,162

0.75 317 44 $24,887 166 145 282 14 $27,268

With two orders retailer buys less, supplier sells less.

Why should the supplier reduce its replenishment lead time?

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Postponement is a cheaper way of providing product variety

Dell delivers customized PC in a few days Electronic products are customized according to their distribution channels Toyota is promising to build cars to customer specifications and deliver

them in a few days Increased product variety makes forecasts for individual products inaccurate

– Lee and Billington (1994) reports 400% forecast errors for high technology products

– Demand supply mismatch is a problem» Huge end-of-the season inventory write-offs. Johnson and Anderson (2000) estimates

the cost of inventory holding in PC business 50% per year. Not providing product flexibility leads to market loss.

– An American tool manufacturer failed to provide product variety and lost market share to a Japanese competitor. Details in McCutcheon et. al. (1994).

Postponement: Delaying the commitment of the work-in-process inventory to a particular product, a.k.a. end of line configuration, late point differentiation, delayed product differentiation.

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Postponement

Postponement is delaying customization step as much as possible

Need:– Indistinguishable products before customization

– Customization step is high value added

– Unpredictable demand

– Negatively correlated product demands

– Flexible SC to allow for any choice of customization step

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Forms of Postponement by Zinn and Bowersox (1988)

Labeling postponement: Standard product is labeled differently based on the realized demand.– HP printer division places labels in appropriate language on to printers after the

demand is observed.

Packaging postponement: Packaging performed at the distribution center. – In electronics manufacturing, semi-finished goods are transported from SE Asia to

North America and Europe where they are localized according to local language and

power supply

Assembly and manufacturing postponement: Assembly or manufacturing is done after observing the demand.– McDonalds assembles meal menus after customer order.

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Examples of Postponement HP DeskJet Printers

– Printers localized with power supply module, power cord terminators, manuals Assembly of IBM RS/6000 Server

– 50-75 end products differentiated by 10 features or components. Assembly used to start from scratch after customer order. Takes too long.

– Instead IBM stocks semi finished RS/6000 called vanilla boxes. Vanilla boxes are customized according to customer specification.

Xilinx Integrated Circuits– Semi-finished products, called dies, are held in the inventory. For easily/fast customizable

products, customization starts from dies and no finished goods inventory is held. For more complicated products finished goods inventory is held and is supplied from the dies inventory.

– New programmable logic devices which can be customized by the customer using a specific software.

Motorola cell phones– Distribution centers have the cell phones, phone service provider logos and service provider

literature. The product is customized for different service providers after demand is materialized.

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Postponement Saves Inventory holding cost by reducing safety stock

– Inventory pooling

– Resolution of uncertainty

Saves Obsolescence cost Increases Sales Stretches the Supply Chain

– Suppliers

– Production facilities, redesigns for component commonality

– Warehouses

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Value of Postponement: Benetton case

For each color, 20 weeks in advance forecasts– Mean demand= 1,000; Standard Deviation= 500

For each garment– Sale price = $50– Salvage value = $10– Production cost using option 1 (long lead time) = $20

» Dye the thread and then knit the garment

– Production cost using option 2 (short lead time) = $22» Knit the garment and then dye the garment

What is the value of postponement?– p=50; s=10; c=20 or c=22

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Value of Postponement: Benetton case

CSL=(p-c)/(p-c+c-s)=30/40=0.75 Q=norminv(0.75,1000,500)=1,337

00,1))337,1000,5normdist(1(1 1337 20)(50

00,1)337,1000,5normdist(1 1337 10)-(20 -

0,1,0)1000)/500,-1337normdist(( 500 10)-(50 -

0,1,1)1000)/500,-1337normdist(( 1000 10)-(50 Profit Expected

Expected profit by using option 1 for all products4 x 23,644=$94,576

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Apply option 2 to all products: Benetton case

CSL=(p-c)/(p-c+c-s)=28/40=0.70 Demand is normal with mean 4 x 1000 and st.dev sqrt(4) x 500 Q=norminv(0.75,4000,1000)=4524

000,1))524,4000,1normdist(4(1 4524 22)(50

000,1)524,4000,1normdist(4 4524 10)-(22 -

,0,1,0)4000)/1000-4524normdist(( 1000 10)-(50 -

,0,1,1)4000)/1000-4524normdist(( 4000 10)-(50 Profit Expected

Expected profit by using option 2 for all products=$98,902

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Postponement Downside

By postponing all three garment types, production cost of each product goes up

When this increase is substantial or a single product’s demand dominates all other’s (causing limited uncertainty reduction via aggregation), a partial postponement scheme is preferable to full postponement.

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Partial Postponement: Dominating Demand Color with dominant demand: Mean = 3,100, SD = 800 Other three colors: Mean = 300, SD = 200

Expected profit without postponement = $102,205 Expected profit with postponement = $99,872

Are these cases comparable?

– Total expected demand is the same=4000

– Total variance originally = 4*250,000=1,000,000

– Total variance now=800*800+3(200*200)=640,000+120,00=760,000

Dominating demand yields less profit even with less total variance. Postponement can not be any better with more variance.

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Partial Postponement: Benetton case

For each product a part of the demand is aggregated, the rest is not Produce Q1 units for each color using Option 1 and QA units

(aggregate) using Option 2, results from simulation:

Q1 for each QA Profit

1337 0 $94,576

0 4524 $98,092

1100 550 $99,180

1000 850 $100,312

800 1550 $104,603

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Tailored (Dual) Sourcing

Tailored sourcing does not mean buying from two arbitrary sources. These two sources must be complementary: – Primary source: Low cost, long lead time supplier

» Cost = $245, Lead time = 9 weeks

– Complementary source: High cost, short lead time supplier» Cost = $250, Lead time = 1 week

An example CWP (Crafted With Pride) of apparel industry bringing out competitive advantages of buying from domestic suppliers vs international suppliers.

Another example is Benetton’s practice of using international suppliers as primary and domestic (Italian) suppliers as complementary sources.

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Tailored Sourcing: Multiple Sourcing Sites

Characteristic Complementary Site Primary Site

Manufacturing Cost

High Low

Flexibility (Volume/Mix)

High Low

Responsiveness High Low

Engineering Support

High Low

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Dual Sourcing Strategies from the Semiconductor Industry

Strategy Complementary Site Primary Site

Volume based dual sourcing

Fluctuation Stable demand

Product based dual sourcing

Unpredictable products, Small

batch

Predictable, large batch products

Model based dual sourcing

Newer products Older stable products

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Learning Objectives Optimal order quantities are obtained by

trading off cost of lost sales and cost of excess stock

Levers for improving profitability– Increase salvage value and decrease cost of stockout

– Improved forecasting

– Quick response with multiple orders

– Postponement

– Tailored sourcing